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arXiv · 1606.02445

4D Pyritohedral Symmetry with Quaternions, Related Polytopes and Lattices

Mehmet Koca·Nazife Ozdes Koca·Amal Juma Hamood Al-Qanobi
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Abstract

We describe extension of the pyritohedral symmetry to 4-dimensional Euclidean space and present the group elements in terms of quaternions. It turns out that it is a maximal subgroup of both the rank-4 Coxeter groups W(F4) and W(H4) implying that it is a group relevant to the crystallography as well as quasicrystallographic structures in 4-dimensions. First we review the pyritohedral symmetry in 3 dimensional Euclidean space which is a maximal subgroup both in the Coxeter-Weyl groups W(B3)=Aut(D3) and W(H3). The related polyhedra in 3-dimensions are the two dual polyhedra pseudoicosahedron- pyritohedron and the pseudo icosidodecahedron. In quaternionic representations it finds a natural extension to the 4-dimensions.The related polytopes turn out to be the pseudo snub 24-cell and its dual polytope expressed in terms of a parameter x leading to snub 24-cell and its dual in the limit where the parameter x takes the golden ratio. It turns out that the relevant lattice is the root lattice of W(D4).

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Mehmet Koca, Nazife Ozdes Koca, Amal Juma Hamood Al-Qanobi. 2016-06-08. 4D Pyritohedral Symmetry with Quaternions, Related Polytopes and Lattices. https://doi.org/10.24200/squjs.vol21iss2pp150-161

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